Theorem dcomex 10357

Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
dcomex	⊢ ω ∈ V

Proof of Theorem dcomex

Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 8415	. . . . . . 7 ⊢ 1o ≠ ∅
2		df-br 5099	. . . . . . . 8 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3		elsni 4597	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4		fvex 6847	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ∈ V
5		fvex 6847	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑛) ∈ V
6	4, 5	opth1 5423	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓‘𝑛) = 1o)
7	3, 6	syl 17	. . . . . . . 8 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓‘𝑛) = 1o)
8	2, 7	sylbi 217	. . . . . . 7 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓‘𝑛) = 1o)
9		tz6.12i 6860	. . . . . . 7 ⊢ (1o ≠ ∅ → ((𝑓‘𝑛) = 1o → 𝑛𝑓1o))
10	1, 8, 9	mpsyl 68	. . . . . 6 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11		vex 3444	. . . . . . 7 ⊢ 𝑛 ∈ V
12		1oex 8407	. . . . . . 7 ⊢ 1o ∈ V
13	11, 12	breldm 5857	. . . . . 6 ⊢ (𝑛𝑓1o → 𝑛 ∈ dom 𝑓)
14	10, 13	syl 17	. . . . 5 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
15	14	ralimi 3073	. . . 4 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16		dfss3 3922	. . . 4 ⊢ (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17	15, 16	sylibr 234	. . 3 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18		vex 3444	. . . . 5 ⊢ 𝑓 ∈ V
19	18	dmex 7851	. . . 4 ⊢ dom 𝑓 ∈ V
20	19	ssex 5266	. . 3 ⊢ (ω ⊆ dom 𝑓 → ω ∈ V)
21	17, 20	syl 17	. 2 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22		snex 5381	. . 3 ⊢ {⟨1o, 1o⟩} ∈ V
23	12, 12	fvsn 7127	. . . . . . . 8 ⊢ ({⟨1o, 1o⟩}‘1o) = 1o
24	12, 12	funsn 6545	. . . . . . . . 9 ⊢ Fun {⟨1o, 1o⟩}
25	12	snid 4619	. . . . . . . . . 10 ⊢ 1o ∈ {1o}
26	12	dmsnop 6174	. . . . . . . . . 10 ⊢ dom {⟨1o, 1o⟩} = {1o}
27	25, 26	eleqtrri 2835	. . . . . . . . 9 ⊢ 1o ∈ dom {⟨1o, 1o⟩}
28		funbrfvb 6887	. . . . . . . . 9 ⊢ ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
29	24, 27, 28	mp2an 692	. . . . . . . 8 ⊢ (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
30	23, 29	mpbi 230	. . . . . . 7 ⊢ 1o{⟨1o, 1o⟩}1o
31		breq12 5103	. . . . . . . 8 ⊢ ((𝑠 = 1o ∧ 𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
32	12, 12, 31	spc2ev 3561	. . . . . . 7 ⊢ (1o{⟨1o, 1o⟩}1o → ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡)
33	30, 32	ax-mp 5	. . . . . 6 ⊢ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡
34		breq 5100	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡 ↔ 𝑠{⟨1o, 1o⟩}𝑡))
35	34	2exbidv 1925	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑠∃𝑡 𝑠𝑥𝑡 ↔ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡))
36	33, 35	mpbiri 258	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ∃𝑠∃𝑡 𝑠𝑥𝑡)
37		ssid 3956	. . . . . . 7 ⊢ {1o} ⊆ {1o}
38	12	rnsnop 6182	. . . . . . 7 ⊢ ran {⟨1o, 1o⟩} = {1o}
39	37, 38, 26	3sstr4i 3985	. . . . . 6 ⊢ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40		rneq 5885	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41		dmeq 5852	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
42	40, 41	sseq12d 3967	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
43	39, 42	mpbiri 258	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44		pm5.5 361	. . . . 5 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
45	36, 43, 44	syl2anc 584	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
46		breq 5100	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
47	46	ralbidv 3159	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
48	47	exbidv 1922	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
49	45, 48	bitrd 279	. . 3 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50		ax-dc 10356	. . 3 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
51	22, 49, 50	vtocl 3515	. 2 ⊢ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
52	21, 51	exlimiiv 1932	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  {csn 4580  ⟨cop 4586   class class class wbr 5098  dom cdm 5624  ran crn 5625  suc csuc 6319  Fun wfun 6486  ‘cfv 6492  ωcom 7808  1_oc1o 8390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-dc 10356

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-1o 8397

This theorem is referenced by:  axdc2lem  10358  axdc3lem  10360  axdc4lem  10365  axcclem  10367  precsexlem10  28212  seqsex  28281  noseqex  28285